By a blackbox theorem I mean a theorem that is often applied but whose proof is understood in detail by relatively few of those who use it. A prototypical example is the Classification of Finite Simple Groups (assuming the proof is complete). I think very few people really know the nuts and bolts of the proof but it is widely applied in many areas of mathematics. I would prefer not to include as a blackbox theorem exotic counterexamples because they are not usually applied in the same sense as the Classification of Finite Simple Groups.

I am curious to compile a list of such blackbox theorems with the usual CW rules of one example per answer.

Obviously this is not connected to my research directly so I can understand if this gets closed.

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Domain of use is important here. Many theorems are invoked by physicists who have no idea of the actual proofs.
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Steve HuntsmanJun 13 '12 at 22:05

7

@Zsbán: That theorem has nice consequences, e.g., in finite geometry. But treating it as a blackbox is just laziness, since the proof is just a couple of pages of basic graduate algebra.
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Felipe VolochJun 15 '12 at 0:19

2

The classification is used by people working on permutation groups and graph theory all the time. Also they are used in profinite group theory.
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Benjamin SteinbergJun 16 '12 at 19:21

2

In my mind I was hoping for things used in at least 100 papers and understood in all technical detail by fewer than 5% of people in the general area to which the theorem belongs. But it need not be this rigid.
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Benjamin SteinbergJun 17 '12 at 3:08

60 Answers
60

The graph minor theorem and the graph structure theorem are two results which are invoked quite often in combinatorics/graph theory. Much like the classification of finite simple groups they are excellent ways of sweeping hundreds of pages of technical proofs under just a few sentences.

There are many papers by H. Hauser whose message is "You can understand Hironaka's proof!". There is even a game-theoretic interpretation. Very recommended.
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Martin BrandenburgJun 14 '12 at 8:15

2

There are also now books (one by Kollar and another by Cutkosky) that aim to present the proof at a graduate student level. I think they don't prove the most general/detailed statements from Hironaka's original paper, though.
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Dan RamrasJun 15 '12 at 0:34

Deligne's Theorem, found at Wikipedia under the heading of Weil conjectures, which is the Riemann Hypothesis for zeta-functions of algebraic varieties over finite fields, is often applied to estimate exponential sums in Number Theory, I suspect often by people (like me) who haven't gone through a proof in detail.

Low dimensional topology is unfortunately full of such theorems. Maybe the archetypal example is the Kirby Theorem, which states that surgery on two framed links in S3 give diffeomorphic 3-manifolds if and only if the links are related by a specific set of combinatorial moves. The result is used routinely, in order to prove that invariants of framed links descend to topological invariants of the manifold (e.g. Reshetikhin-Turaev invariants).

All known proofs of Kirby's Theorem are a nightmare (see this MO question). You need to use some heavy tool (Cerf's Theorem/ explicit presentation of Mapping Class Groups) in order to show that some expansion of the space of Morse functions (a Frechet space) is path connected. This is outside the toolbox of most topologists.

I would be surprised if there were 20 people in the world who have read through and understood the details of the proof of Kirby's Theorem. Yet it's routinely used.

There are more mild examples too. The proof that PL 3-manifolds can be smoothed, and that the resulting smooth structure is unique up to isotopy (the exact statement is in Kirby-Seibenmann), is used routinely as though it were obvious, but it is actually quite a hard theorem which is not covered in any of the standard textbooks (Thurston's "3-Manifolds" being an exception). See Lurie's 2009 notes.

Freedman's theorem "Casson handles are handles" is also used as a black box by many people. Once this is known, standard arguments from higher dimensions can be pushed down to 4 dimensions to prove h-cobordism and Poincare. Hopefully this will be rectified next summer when an extended workshop will go over the proof (I think at Bonn).
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Ian AgolJun 14 '12 at 17:20

1

I confess that I have no looked at the proof of the Kirby calculus theorem either recently. But I personally think that the difficulty of the Thom transversality theorem and Cerf theory are overplayed. Is the Reidemeister move theorem for smooth knots a difficult theorem? It's the same sort of thing. Yes, there are a lot of details if you want to be very rigorous, but the lemmas all have natural statements. For instance, you can prove Thom transversality in the setting of a finite-dimensional vector space of polynomial functions, using algebraic geometry.
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Greg KuperbergJun 19 '12 at 6:11

I think the Uniformization theorem is an example of blackbox theorem : any simply connected Riemann surface is conformally equivalent to either the open unit disk, the complex plane or the Riemann sphere.

The standard proof of the Uniformization theorem with the Green's function, while rather involved, shouldn't really surpass the ability of most who come across it. There also exists a short and elegant proof that uses certain rather more advanced tools: the Mayer-Vietoris sequence and the celebrated Newlander-Nirenberg theorem. But NN for surfaces is just the existence of isothermal coordinates, which is much simpler to prove. This proof can be found in Demailly's "Complex Analytic and Differential Geometry" (available at www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf).
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HeWhoHungersJun 16 '12 at 1:24

3

@HeWhoHungers: I agree that the proof in Demailly's book is marvellous and elegant, but it is neither easy or short in any sense. I talked about exactly this proof in a lecture course on Teichmüller theory some years ago, addressing an audience of very bright graduate students. I needed 3 or 4 hours to communicate the proof and I remember it to be a tour de force, both for me and the audience. Even if you take the advanced tools for granted (as I did), the details (many of which are thrown under the carpet in the book) are very, very subtle.
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Johannes EbertJun 26 '12 at 17:40

2

The ration #{people who quote the theorem on a daily basis} / #{people who know the details of the proof offhand} is very high, so it is a perfect example of a blackbox theorem.
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Johannes EbertJun 26 '12 at 17:47

Faltings' Theorem, to the effect that a curve of genus greater than 1 over the rationals has only finitely many rational points, is often invoked, I suspect often by people who haven't gone through a proof in detail.

I think also many people treat certain tools in homological algebra this way. For example various facts about spectral sequences and how to use them.

In the spectral sequences example, I feel like many people once learned the background, and then forgot it (perhaps could reconstruct if forced). But regardless, they still know how to apply the machines in the problems relevant to them.

This topology theorem states that a looped continuous path in the plane partitions the points of the plane, such that any continuous path going from a point in one partition to a point in the other intersects the loop.

There seem to be a lot of theorems in calculus of which I don't fully understand the proof, though some of this shows my ignorance of calculus. Jordan's theorem seem to be an extreme example though. Let me list some other examples.

the existance and basic properties of the Lebesgue measure and infinite product measures

the fact that a Wiener process is almost surely everywhere continuous (mentioned below as a separate answer by weakstar)

the fact that the roots of a complex polynomial (or the eigenvalues of a complex matrix) are continuous in the coefficients (though I should learn the proof for this because the more precise statements on how well conditioned the roots are on the coefficients is useful)

the spectral theorem about linear maps on a possibly infinite-dimensional Hilbert-space

the proof that a convex function (from reals to reals) is always continuous everywhere and has a left and right derivative everywhere (Update: okay, remove this last one because Ian Morris gave a simple proof below. I seemed to remember it was more difficult than that. Thanks, Ian.)

Rademacher's theorem: every Lipschitz function from an open subset of $ \mathbb{R}^m $ to $ \mathbb{R}^n $ is differentiable almost everywhere. (Added on Paul Siegel's suggestion. For some reason I haven't heared of this theorem before, but it sure sounds useful.)

Lebesgue's criterium which claims that a bounded function from reals to reals is Riemann-integrable iff it's continuous almost everywhere. (The proof is elementary and doesn't require any ideas, but it's laborous.)

This example is not really what I want because all basic algebraic topology books do it.
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Benjamin SteinbergJun 13 '12 at 22:59

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Most of those are genuinely laborious proofs, but the one about convex functions can be done in a few lines. A convex function clearly has at most two intervals of monotonicity, which implies that the left and right limits at each point exist. If they aren't the same for some point then we can find a chord between two points of the graph close to the discontinuity which passes below the graph (on the left if the jump is downwards, or to the right if it is upwards) contradicting convexity. Differentiability is obtained by showing that (f(x+r)−f(x))/r is monotone in r.
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Ian MorrisJun 14 '12 at 13:12

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Perhaps you could replace your fifth example with Rademacher's theorem: every Lipschitz function from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$ is differentiable almost everywhere. This is a more serious result which people use all the time, and I'm not sure everyone really knows the proof (though maybe I should speak for myself).
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Paul SiegelJun 14 '12 at 23:39

Existence and uniqueness of invariant
Haar measure on a locally compact
topological group.

It is used in harmonic analysis and number theory. It is not so difficult a result to state but a proof is not so commonly seen in books. The measure allows one to define an integral on the group and do analysis.

What is not terribly well known (or exposited in very many books) is the constructive proof of existence and uniqueness of Haar measure that does not use the axiom of choice. While I imagine the vast majority of people who make use of Haar measure either don't care about the axiom of choice or have nicer constructions as Ben Wieland suggests, it is at very least an interesting curiosity that the axiom of choice is not needed at all, since the usual proof one sees relies so crucially on it.
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Evan JenkinsJun 15 '12 at 17:44

1

@EvanJenkins: do you have a reference for the non-AC proof? When studying Haar measure construction I found a lot of texts doing only the compact case, and one text with an AC proof of the locally compact case.
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Emilio PisantyJun 28 '12 at 11:56

Nagata embedding is another black box - its statement is very simple and useful, but its proof is hard.

By combining Nagata embedding with Hironaka's resolution of singularities (mentioned in another answer), you get "any smooth variety over a characteristic zero field admits an open immersion into a proper smooth variety", which is concise enough that people often use it without citing the authors' hard work.

The existence of Hilbert and Quot schemes. These are arguably the most important objects in moduli/deformation theory but the proof of their existence is almost never even presented in books on the topic, let alone needed or used. All the properties and applications follow formally so the existence is used as a black box.

The decomposition theorem for perverse sheaves is used in many areas of mathematics, for example representation theory, while the details of the weights machinery involved in its proofs are notoriously hard.

Someone mentioned existence and uniqueness of Haar measure on a locally compact topological group. But if one uses the Riesz representation theorem and Tychonoff, the standard proof is not so long or hard, and may even be considered conceptual. For example a clear proof is in Bourbaki's Integration, and in Principles of Harmonic Analysis [by Deitmar and Echterhoff].

is more often used as a Blackbox theorem. Of course this is a main result in analysis, and many standard books (Rudin, Folland, Appendix of Conway's Functional analysis) have a proof, but they are all long and technical, and in my opinion very difficult to remember. See also Remark 4 in these wonderful notes by Terry Tao.

When learning algebraic geometry and in particular the notion of smooth varieties, you will probably stumble upon the following Theorems:

Regular local rings are factorial.

Localizations of regular local rings are regular, too.

A local ring is regular iff its residue field has finite projective dimension (Serre).

Many texts on algebraic geometry take this as a black box, quoting standard sources of commutative algebra. The reason seems to be that you don't have to understand the methods of the proof (e.g. Koszul homology) in order to apply these results.

My immediate thought upon seeing the question, and, I believe, one of the biggest examples of this phenomenon, is:

Class Field Theory

Almost anyone working in algebraic number theory uses the main results of class field theory regularly. However, even if many people have sat through a course going through the proofs of the theorems, very few people remember the proofs, and even fewer use them.

I recall hearing advice from various mathematicians that the most important thing is to learn the statements of class field theory, but not the proofs.

While it is somewhat instructive to
know what goes into the proofs of the
main theorems (e.g., to see what
obstacles prevent the proofs from
being entirely constructive), it
cannot be said that the grungy details
of these proofs are particularly
relevant to using the theory in
practice. Thus, in the first half of
the course we will emphasize an
understanding of the statements of the
main results (in their many different
forms) and will not place much
emphasis on how the main theorems are
proven; precise references will be
given for those who wish to read the
details of the proofs of the main
theorems. Once we have spent some time
digesting what class field theory
tells us, we will study some
applications of the theory, such as in
the context of imaginary quadratic
fields and abelian coverings of
algebraic curves.

Faltings' almost purity theorem. The proof given, for the smooth case, in $p$-adic Hodge theory has some problems, and the proof of the general case in the Asterisque paper Almost Étale Extensions is completely unreadable (at least to me) and also contains some mistakes. We now finally have a very good proof (by Peter Scholze), but the almost purity theorem has been used as a black box for years.

I think differential topology has dozens of these results. Here are some examples that immediately come to mind:

The tubular neighborhood theorem: every submanifold $N$ of a manifold $M$ has an open neighborhood which is diffeomorphic to the total space of the normal bundle of $N$

The fundamental theorem of Morse theory: if $f: M \to \mathbb{R}$ is a Morse function and $[a,b]$ is an interval which contains no critical values of $f$ then the set of all points where $f \leq a$ is a deformation retract of the set of all points where $f \leq b$

Every continuous isomorphism of smooth vector bundles is homotopic to a smooth isomorphism (and other such "continuous equivalence = smooth equivalence" results)

Probably most topologists know the basic ideas behind the proofs of these results, but I think many would be hard-pressed to actually write down a complete argument. I say this with confidence because I know of several textbooks by good authors that have proofs which are either wrong or sketchy on some details.

There are also some results with standard proofs that are widely known, but I think considerably more people use the results than know the proofs:

De Rham's theorem: the De Rham cohomology groups of a manifold are isomorphic to the singular cohomology groups with real coefficients

The Hodge theorem: every De Rham cohomology class on a Riemannian manifold has a harmonic representative

Whitehead's result that every smooth manifold has a unique PL structure

This doesn't quite seem to be in the spirit of what the original questioner was going for. This is more like, "Mathematical facts that people think that they understand but are actually a bit trickier than they think," which is interesting, but perhaps deserves its own thread.
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Dan LeeJun 15 '12 at 20:36

Is this theorem actually applied frequently inside or outside of set theory?
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Jan WeidnerJun 13 '12 at 21:59

4

Would you use AC if it contradicts ZF ? The magnitude of the independence theorem is that we use it implicitely whenever we apply AC, since it tells us that AC doesn't lead to logical contradictions that weren't already present in ZF.
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RalphJun 13 '12 at 22:20

12

@Ralph, I disagree. First, the independence of AC from ZF is not the same as the consistency of ZFC relative to ZF (indeed, the latter is very easy). Second, I'm not certain that even this is used frequently outside of set theory; when non-logicians use the axiom of choice, they are not tacitly assuming that it is consistent with ZF, they are tacitly assuming that it is part of some consistent set theory - for example, how many non-logicians know the ZF axioms off the top of their head? I think in practice the set theory that is actually used is generally some high-but-finite-order arithmetic.
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Noah SJun 13 '12 at 22:56

1

@Ralph: you may say the same about any axiom in any theory.
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Michal R. PrzybylekJun 13 '12 at 23:41

1

In fact, building off of Michal, perhaps the consistency of the axioms of powerset, replacement, and separation would be better, since these are implicitly used whenever comprehension (forming the set of all $x$ such that $P(x)$) is used, and full comprehension actually is contradictory! But I still don't feel that these are good examples. Roughly speaking, either you're Platonist - in which case mere consistency of AC isn't sufficient to justify using it - or one is interested in proving theorems from axioms, in which case "ZFC proves X" is valuable even if ZFC isn't known to be consistent.
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Noah SJun 14 '12 at 0:24

@Zsbán: Continuity of BM is part of the standard definition, so proving that is the same as proving that it exists. However, the proof that BM is almost-surely nowhere differentiable is probably less well known.
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George LowtherJun 14 '12 at 22:03

1

The proof of continuity usually follows from the "Kolmogorov Criterion": If there exists strictly positive constants $\varepsilon$, $p$ and $C$ such that $$\mathbb{E}|X_t - X_s|^p \leq C|t-s|^{1+\varepsilon}$$ then almost surely $X$ has a modification which has $\alpha$-Hölder continuous paths for any $\alpha \in (0,\frac{\varepsilon}{p})$
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Felipe OlmosJun 15 '12 at 23:54

(A hamiltonian graph is a graph that has a cycle passing through every node.) Everyone likes to use this theorem for proving other NP-completeness proofs, but few people would know an actual proof. Even the simplest proof is somewhat messy. The theorem that 3-colorable graphs are NP-complete is similar.

For a long time, the Littlewood-Richardson rule has been a black box. See van Leeuwen's wonderful article for its history (and a rather involved, even if enlightening proof). This really changed with Stembridge's 2-pages long slick (although far from straightforward!) proof (2002) and Gasharov's 3-pages long proof (1998). (I have read Stembridge and can vouch for its good exposition; it's not short by virtue of being unreadable, but short by virtue of being short. I have not yet read Gasharov, and I am in the middle of van Leeuwen.)

Voiculescu's theorem: an ample representation of a C*-algebra essentially absorbs any nondegenerate representation

Kasparov's technical theorem: if anybody really cares, I'll repeat the statement. The point is that anybody who works with bivariant K-theory uses this result ALL THE TIME, e.g. for excision or the existence of Kasparov products.

Stinespring's theorem: any completely positive map into $B(H)$ dilates to a representation

I have been using Voicalescu's theorem and KTT for a about a year or so longer than I knew the proofs. I probably still wouldn't know the proofs if it hadn't become necessary. Stinespring's theorem is probably better known among the people who use it because it's not so difficult, but it could be tempting to use it as a black box.

Nice answers. An addendum/afterthought: it probably isn't used very often by practising operator algebraists, but the equivalence of nuclearity and amenability for C*-algebras gets invoked a lot of the time by people in Banach algebras, and I rather doubt many of them have actually worked through all the details.
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Yemon ChoiJun 15 '12 at 0:26